We are going to show that \(1\) is
the largest natural number.
Proof: Let \(n\) be the largest natural
number. We will show that \(n=1\).
Since \(n\) is the largest natural
number, \(n^2\), which is also a
natural number, must be less than or equal to \(n\). Therefore \(n^2-n\leq 0\). But \(n^2-n=n(n-1)\). If \(n(n-1)\leq 0\), it must be that \(0\leq n\leq 1\), and so \(n=1\).